Product formulas for multiple stochastic integrals associated with L\'evy processes

TL;DR

This paper derives explicit product formulas for multiple stochastic integrals with respect to Lévy processes, including jump and Gaussian parts, and applies these to moments, cumulants, and a central limit theorem.

Contribution

It introduces new product formulas for multiple integrals involving Lévy processes with jumps, extending known results for Brownian motion.

Findings

Explicit formulas for expectations of products of iterated integrals

Derived moments and cumulants for stochastic integrals w.r.t. Lévy measures

Established a central limit theorem for long-term behavior

Abstract

In the present paper, we obtain an explicit product formula for products of multiple integrals w.r.t. a random measure associated with a L\'evy process. As a building block, we use a representation formula for products of martingales from a compensated-covariation stable family. This enables us to consider L\'evy processes with both jump and Gaussian part. It is well known that for multiple integrals w.r.t. the Brownian motion such product formulas exist without further integrability conditions on the kernels. However, if a jump part is present, this is, in general, false. Therefore, we provide here sufficient conditions on the kernels which allow us to establish product formulas. As an application, we obtain explicit expressions for the expectation of products of iterated integrals, as well as for the moments and the cumulants for stochastic integrals w.r.t. the random measure. Based on these expressions, we show a central limit theorem for the long time behaviour of a class of stochastic integrals. Finally, we provide methods to calculate the number of summands in the product formula.